Kinetics of desorption from a heterogeneous surface

PHYSICAL REVIEW BVOLUME 26, NUMBER 8 Kinetics of desorption from aheterogeneous surface 15 OCTOBER 1982 A. Cordoba and J.J.Luque Departamento de Termologia, Facultad de Fisica, Universidad de Sevilla, Sevilla, Spain (Received 9April 1982;revised manuscript received 26 July 1982) Mathematical equations governing kinetics of desorption from heterogeneous surfaces are derived from amaster equation with the assumption that only nearest-neighbor adatoms interact. Anumber of cases are analyzed: random, periodical, and patchwise distributions of sites characterized by different activation energies. In general, the kinetic equations obtained must be solved numerically. We have performed numerical calculations for several specific cases and analyzed the influence exerted by heterogeneity and lateral interaction between adatoms on the desorption rate d8/dT. The results obtained make it clear that, due to lateral interaction, the desorption curves are very sensitive to the way in which heterogeneities are distributed on the surface (periodically, patches with different sizes, etc.). I. INTRODUCTION In general, thermal desorption of agas from a heterogeneous surface gives rise to very different desorption curves from those resulting for desorption from ahomogeneous surface. Heterogeneity can be caused by the adsorbent surface or by lateral interaction between adsorbed particles. Fractional, zero, or very high-order desorption kinetics can result from afirst-order desorption mechanism with an activation energy and apreexponential factor that depend on the kind of lattice site where the adatom desorption occurs, or from the above mentioned lateral interaction. The simplest method in dealing with heterogeneity is to assume an activation energy which is afunction of coverage. A variety of empirical functions have been used' and they have been useful in many cases. However, this method does not allow us to gain insight into the various mechanisms that make activation energy dependent on coverage. Our purpose in this paper is to study the way in which substrate heterogeneity and lateral interaction between adatoms influence the desorption process. We start from amaster equation, where transition probabilities are taken in the Arrhenius form. We consider two contributions to activation energy, namely, the energy due to the substrate and that due to lateral interaction between nearest-neighbor adatoms on the surface. We have carried out an analysis of the desorption process for avariety of conditions. First we consider alinear chain where there are sites characterized with different activation energies due to the substrate and governed by arandom distribution. The lateral interaction energy is agiven constant and the Bragg-William approximation is used. Then we extend the formulation to treat lattices with any coordination number c. Later we study the case where the sites with the same activation energy are distributed periodically (periodical heterogeneous chain) and, subsequently, we consider arandom patchwise model. Finally, we discuss our results and summarize our conclusions. II. RANDOM SITE DISTRIBUTION Firstly, we consider alinear chain with Esites (N— +oo )where each site can be empty or occupied by one adatom. We associate avariable s& with each chain site, which takes the values 1(filled site) or — 1(empty site). Astate of the chain can be described by aset Isj I(j=1,...,N) and we denote P(s~, ...,sN, t) the probability that the system is found in configuration Is~, ...,'s~j at time t. Two terms contribute to the activation energy, namely, anearest-neighbor interaction energy ho, given as constant, and an energy due to the substrate causing the heterogeneity. For simplicity we develop the model according to the case in which there are two types of sites, called 1and 2, which contribute to the activation energy with energies h~ and h2, respectively. Such amodel is similar to a "heterogeneous" Ising model. We assume that only the desorption mechanism is operative, and readsorption onto the chain and migration of adatoms from site to site on the chain are inhibited. When aparticle is desorbed from the jth site, the value of sj changes from 1to — 1. The time dependence of P(s~, ...,sz,t) can be described by the master equation 26 1982 The American Physical Society KINETICS OF DESORPTION FROM AHETEROGENEOUS. .. dP($ ], .. .&sp. ..,$N', t) dt W(sj ]&sj psj+]) P(s], ...,sp. . .,Sh]', t) j +g[F(sj ]p sj &sj+])P($]p &Sj&, ,p$]v&t)] j where W(sj ],sj,sj+]), which depends on sj ], sj, and st+] because of nearesteighbor interaction, is the transition probability per unit time from state sj to state — sj, while the other sk (kQj) variables remain temporarily fixed. These transition probabilities are taken in the Arrhenius form. There are three different probabilities, as shown in Fig. 1, W'++ — — A;exp[ — (h +2hp)/k]] T], W'+ — — W' +— — A;exp[ — (h;+hp)/k]]T], W' =A;exp( h; /kj— ]T), (2) where superscript i(i =1,2) indicates whether the central site is of type 1or of type 2, A; is the preexponential factor, k]] is the Boltzmann constant, and Tis the temperature. We define the following correlation functions: m z,„=—ys, ,), J 1km Skm =gsj — ]sj j 1kn r,.„= y*, ;,"„), — m mn Ukmn +~Sj ~Sj Sj J where (f(sk) }=gf(sk)P(Isj J;t) Is and the sum gj (i =1,2) extends over all chain sites of type iAlso, w.hen suitable for clarity, we have written sj (i =1,2) instead of sj to indicate that the jth site is of type i From Eqs. (1)and (2) we obtain the kinetic equations dQ =— —,[n;a'] +a']Q;+2a 2(R ];+R2;)+2a2(S];+S2;) +a3(T];]+2T];i+T2;2)+a3(U];]+2U]]2+U2;2)], i=1,2 where n; indicates the fraction of sites of type iin the chain (n]+n2 — — 1) and rA; h; Ik]]T( hplk]] T— — iAh(lk]]T( 2hplksT 1— )— l '=A — h; Ik]] T( hplkgT— l The adsorption degree on sites of type iis n;+Q; l2 and the total adsorption degree is e=ge, . Because Eqs. (5) involve correlation functions comprising more than one chain site, we need to ap- 4030 A. CORDOBA AND J.J.LUQUE 26 I+I+I+I =I+I-I+I 'NyI+I+I-I =I+I-I-I I-I+I+I =I-I-I+I W Now it is very easy to extend the above equations for the case of ndifferent types of sites and for a continuous distribution of types of sites (see the Appendix). When the chain is homogeneous, we find W—— I-I+I-I =I-I-I-I FIG. 1. Schematic diagram of the removal probabilities. =— Ae [1+(e '— 1}8] dt (13) ply some method of approximation to solve these equations. The simplest method is the BraggWilliam approximation. More refined approximations can be carried out, but then calculations become more cumbersome. Here we shall apply the Bragg-William approximation. If i,j,kdenote the plus or minus sign (plus and minus signs indicate filled and empty sites, respectively) and m, n,ptake the values 1and 2(1 indicates site of type 1; 2indicates site of type 2), let NJ" be the fraction of nearest site pairs, the first being asite min astate iand the second asite nin astate j(for example, N' +denotes the fraction of site pairs, the first being an empty site of type 1and the second afilled site of type 2), and let NgP be the fraction of triplets of sites of types m, n,pin states i,j,k, respectively. We assume that From Eqs. (11)— (13) it is apparent that the effect of nearest-neighbor interaction is involved only in the factor exp( h/kit T — )8))1— 8 (strongly repulsive interaction), we can write Eq. (13}as d6 3 dt =— Aexp[ — (h +2h0)/kz T]8 (14) (third-order kinetics), which is an equation similar to the Arrhenius equation with E„,=h+2ho — 2k~ Tln9 . [1 ('— 1)8] If ho —— 0, this factor equals 1and Eq. (13) becomes the standard Arrhenius equation. If Ntjk" =NI NJ"Nf, (10) If exp( — ho/k&T) «1 (strongly attractive interaction), Eq. (13)becomes )& [I+(e '— 1)8], i=1,2. For the total adsorption degree the result is a8 (A — h~/keT8 Ah2/keT8dt X[1+(e ''— 1)8]'. (12) ¹being the fraction of sites of type min state i With this assumption, and by means of a straightforward but tedious calculation, Eqs. (5) become de, d8 2 dt =— Aexp( — h/k T)8(1— 8) (15) ~r exP( g/ke T)—(16) where r=A2/A iand g=h2 — hi. Then the adsorption degree must verify the following equations: Note that while in Eq. (14) d8/dt is proportional to (N+/N) (i.e.,to N+++/N in the Br'agg-William approximation), in Eq. (15) it is proportional to (N+/N)(N /N) (i.e.,to N+/N in the BraggWilliam approximation}. Turning now to the general problem, let us consider the heterogeneous case. By means of a straightforward calculation we obtain 8=8,+n (8,/n ) dt (18) 26 KINETICS OF DESORPTION FROM AHETEROGENEOUS. ..4031 The time evolution of 8can be found from Eqs. (17) and (18), at least formally, by elimination of 81. In general, these equations require anumerical solution. If we consider now that adatoms are desorbed from alattice with coordination number cand we apply the Bragg-William approximation, we obtain de) Iggk~T '= — Ae ''e dt X[1+(e '— 1)8]', i=1,2. (19) Equations (19) and (11) are similar, except for the exponent of the factor due to interaction between nearest neighbors. III. PERIODICAL AND PATCHWISE HETEROGENEOUS SURFACES The above analysis is valid for the case where the distribution of different types of sites is purely random. We consider now the case where this distribution is not purely random, but where we have additional information about it. The simplest case is that of aperiodical heterogeneous lattice, i.e.,there are different types of sites but they are distributed in aperiodical form. Let two types of sites, 1and 2, lie in alinear chain. We assume that the densities of each type of site n, and n2, those of duplets n;J (i, j=1,2), and those of triplets n,jk (ij,k=1,2) are known. In Fig. 2, two perodical heterogeneous linear chains 1 are shown. For case (a) n& n2 —— —,,nt~ — — n22 — —— 0— , 11 ~12 21 2n121 212 2~111 ~112 211 2 =n22~ —— n~22 — — n222 — — 0. For case (b) n, =—,, I1 ~2 3~22 0~11 n12 ~21 ~112 ~121 1 ~211 n111 ~212 ~221 122 ~222 When all sites 1are clustered on one part and all sites 2on another, n11 —— n1, n22 — — n2, n12 —— n21 — — 0. For the case of random distribution previously con22 sldered, n11 — — n1, n22 — — n2, n12 — — n21 — — 51n2, 3 n111 — — n1, and so on. Approximation given by Eqs. (9) and (10) may be expressed as grim lm &&ig =&tm%qj ~rlmn lmn 'i)k =lmn%9J' 9'k (20) (21) where qk denotes the relative fraction of sites of type n(n =1,2) which are in the state k (k— ++,— ). Then Eqs. (5) and (6), jointly with Eqs. (20) and (21),lead to d8i h;IksT- =— A;e 'e; dt 2z e, z' X1+— gn;, +— gn;;k Pl~ Elk i,j,k=1,2(22) where z=exp( iIp/ks — T)1. — Finally, we treat the case where there are different types of sites with apatchwise distribution. For the sake of simplicity we consider again two types of sites, 1and 2, with densities n1 and n2, respectively, on alinear chain with Nsites. Patches of type 1and of type 2must be alternatively placed on this chain, ddenoting the number of patches of type 1(or of type 2). Let F~(m) and F2(m) be the respective distribution functions of the patch sizes, i.e.,F;(m) is the probability of finding apatch formed by mconsecutive sites of type i. Functions F;(m) verify and gF;(m)=1 (23) Pf1 M1 7l2 M2 N'(24) where M~ — — QF~(m)m, Mq — — QF2(m)m, (25) M2 2MMN2(26) M; being the average size of type ipatches in the chain. Densities of site groups involved in Eqs. (22), n;, n,j,nJk (ij,k=1,2) can be written as M1 M1+M I112I112I1I2I1I2 I1 I2 I1 f2 I1 I2 I1 I(a) J1J1I2I1 I1 f2I1 I1 I2 fl I1 f2 f1 I1 I2 f(b) FIG. 2. Periodical heterogeneous linear chains: (a) alternating sites; (b) alternating pairs 1-1 and sites 2. n~~ — — — g(m — 1)F~(m) N d =n, — — QF~(m)=n&—M1 4032 A. CORDOBA AND J.J.LUQUE 26 and so on. Substituting Eqs. (26) into Eqs. (22), one gets the kinetic equations governing the desorption process. Extension of the above formulation to twodimensional lattices is difficult because in order to characterize patches one must specify not only their size, but also their form, and then the problem becomes impracticable because the number of possibilities is enormous. However, if one introduces restrictions on the patch forms or knows the distribution of clusters involving asite and its nearest neighbors, for instance, the cluster for square lattices, it is possible to treat the problem in the same way as for the linear chain, although calculations become cumbersome. IV. RESULTS AND DISCUSSION To analyze the way in which lateral interaction and substrate heterogeneity influence the rate desorption, we have carried out numerical calculations for several specific cases. 'We have assumed a temperature-programmed desorption and alinear relationship between temperature Tand time t, i.e., T=Tp+ut where uis aconstant. We have taken A/a=5&(10 K Firstly, we consider ahomogeneous chain [Eq. (13)] with attractive, zero, and repulsive lateral interaction energy. The results are shown in Fig. 3. The effect of lateral interaction on the curve dB/dT vs Tis clear. Repulsive interaction energy results in the desorption starting at alower tern0.04 perature while occurring more smoothly (i.e.,the curve dB/dT vs Texhibits aless sharp maximum) than when interaction does not exist. On the contrary, attractive interaction energy results in the desorption starting at ahigher temperature while occurring in atemperature range more narrow than in the above cases, and the curve d8/dT vs Texhibits avery sharp maximum. Similar results are again shown in Fig. 4, where we present the curve dB/dT vs Tfor aheterogeneous chain, where two types of sites 1and 2are considered, with n~ =nz — — 0.5, and distributed randomly (n,zn;nz- — , nj~ n;nj— — n~) Het.erogeneity yields two maxima in the curve dB/dT vs T, the maximum at low temperature being the sharpest. Results more interesting than the above ones are obtained when we analyze the influence that the heterogeneity distribution on the chain exerts on the curve de/dT vs T. In Figs. 5and 6we show results obtained for achain with two types of sites, with n~ n2 — — 0— . — 5, for four different site distributions: (a) Regular or periodical chain, like the chain shown in Fig. 2(a); (b) random site distribution, with n;J =n;n, and n;Jq n;n~nq, '(— — c) patchwise site distribution, where we assume that the patch size is governed by aPoisson distribution with average size fixed (for the calculations this average size has been taken as equal to ten sites); (d) distribution in two domains, each domain comprising the sites of type 1or of type 2, respectively. In Fig. 5results for attractive lateral interaction are shown. The curve dB/dT vs Texhibits only one maximum for case (a) and two maxima for the others. The difference between the temperatures corresponding to maxima of dB/dT and the sharpness of these maxima increases as the number of pairs 1— 2, n~2, decreases, case (d) being the extreme case. In Fig. 6results for repulsive lateral interaction are plotted. Now the 0.020.02 0.04 h~=h2=70kJ rnoI — — (K &) (IT 0.01 O.P2 300 400 T(K) 400 T(K) FIG. 3. Desorption rate vs temperature for ahomogeneous chain: (a) hp — — — 5kJmol; (b) hp=0 (c) hp=5 kJ mol FIG. 4. Desorption rate vs temperature for arandom heterogeneous linear chain, with nj — — n2 — — 0.5, h~ — — 60 kJmol ', h2 — — 70 kJmol ', and: (a) hp — — — 5kJmol (b) h=0; (c) h=5kJmol KINETICS OF DESORPTION FROM AHETEROGENEOUS. .. 4033 0.02 (K 1) ~e dT 0.01 &00 500 T(K) FIG. 5. Desorption rate vs temperature for aheterogeneous linear chain with n1 — — n2 — — 0.5and h] — — 60 kJmol ', h2 — — 70 kJmol ', h0 — — 5kJmol '. (a) periodical chain; (b) random chain; (c) patches with average size equal to ten sites; (d) distribution in two domains, each domain comprising the sites of type 1or of type 2, respectively. toms coexist, the desorption curve d8ldT vs Tdepends strongly on the type and fraction of adsorbent sites and also on the distribution of these sites on the chain (periodically, patches with different sizes, etc.). This fact makes it clear that lateral interaction can soften or strengthen substrate heterogeneities and accordingly, in order to clarify adsorption-desorption mechanisms, we recognize the need for characterizing the substrate heterogeneity distribution in the most accurate way possible, independently of lateral interaction between adatoms. APPENDIX We can extend the equations in Sec. II to the case of ndifferent types of sites, n; being the fraction of sites of type ion the chain. Equations (11)and (12) become results are inverted with regard to the above ones. In fact, the maxima are the sharpest and the most distant for case (a) and succesively sharp and distant for case (b). For cases (c) and (d) the difference between the values of the maxima is small and the central part of the curve d8/dT is very smooth and nearly flat. For case (d) the maximum at high temperature is slightly higher than the maximum at low temperature. By comparison of Fig. 5and 6, it can also be seen that for repulsive lateral interaction desorption occurs within alarger temperature range with maxima less sharp than for attractive lateral interactions, as we have indicated above. In conclusion, we can remark that when substrate heterogeneity and lateral interaction between ada- =— A;e '6; dt x[1+(e ''— l)8]', l=1, ...,n d6 "— h,./k T dt X[1+(e '— 1)8] and then )riexp( gilkiiT) (Al) (A2) (A3) (A4) 0.02 (l6 d=l (K"1) 0.01 d6 dt — hg /k~ T &Ane El )riexp( — g,./k& T) 300 400 T(K) x[1+(e ''— 1)8]', (AS) FIG. 6. Desorption rate vs temperature for aheterogeneous linear chain with n~— — n2 —— 0.5and h1— — 60 kJmol ', h2 — — 70 kJmol ', h0 — — — 5kJmol ': (a) Periodical chain; (b) random chain; (c) patches with average size equal to ten sites; (d) distribution in two domains, each domain comprising the sites of type 1or of type 2, respectively. where r; =A;/A~ and g;=h; — h~. If instead of adiscrete distribution of types of sites, we consider acontinuous distribution so that f(h)dh is the fraction of sites with energy comprised between h&+h and h&+h+dh, the sums on the right-hand side of Eqs. (A4) and (AS) must be substituted by the corresponding integrals 4034 A. CORDOBA AND J.J.LUQUE (A6) In this case, we obtain de A(h)e — (h i+h)/kBT — nM/k~ T kgT ~(e — 1)ln"(8i/n i) 8=1Mnn! X[1+(e '— 1)8]'.(A7) To solve Eqs. (A6) and (A7) one needs to know A(h) and f(h). So in the case described by Eqs. (A4) and (AS} as in the case described by Eqs. (A6) and (A7), it is not possible to find an exact solution. Indeed, even for the most simplified cases, the resulting equations require anumerical solution. To illustrate the difficulties appearing in this problem, we consider the following continuous cases: A(h)=A =const, i.e.,r=1,and 0, h(0 f(h}= M', 0&h &M 0, h)M. (A8) de dt Ae hi/kg TB Mln(8i/n i) n) X[1+(e ''— l)8]'. (A9) Because e~/n~ cannot be eliminated between Eq. (AS) and Eq. (A9},we must start from d(8i/n i)—hk ire8Ti holkeT- =— Ae '1+(e — 1) 1— dt n~ — nM/k~ T AT ~eln "(8i/ni) Mnn! In"(8 i/n i) nn! (A10) Then we obtain the value of 8i/n iat time t, truncating the series involved in Eq. (A10) in an order depending on the required accuracy. After we have found 8i/n i, we can substitute it into Eq. (A8) and get the value of 8at time t. iSee, for example, J. G. Dash, Films on Solid Surfaces (Academic, New York, 1975), Chap. 9and references therein; A. W. Adamson, Physical Chemistry of Sur faces (Wiley-lnterscience, New York, 1976), Chap. XIV and references therein. B. Kindl, R. A. Pachovsky, B. A. Spencer, and B. W. Wojciechowsky, J. Chem. Soc. Faraday Trans. 169, 1162 (1973). Y. Tokoro, T. Uchijima, and Y. Yoneda, J. Catal. 56, 110(1979). 4See, for example, K. Huang, Statistical Mechanics (Wiley, New York, 1963),Chap. 16. 5I. Morgensten, K. Binder, and A. Baumgartner, J. Chem. Phys. 69, 253 (1978) and references therein; I. Morgensten, K. Binder, and R. M. Hornreich, Phys. Rev. B23, 287 (1981)and references therein. 6G. Schwarz, Ber. Bunsenges. Phys. Chem. 75, 40 (1971);J.Theor. Biol. 36, 569 (1972). 7H. W. Huang, Phys. Rev. A8, 2553 (1973). Y. Saito and R. Kubo, J.Stat. Phys. 15, 233 {1976). J. J. Luque and A. Cordoba, J. Chem. Phys. 76, 6393 (1982).